Finite Group Databasebeta

← #36363701

Subgroups — #36363701

Order 22144.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012025C211yesmaximal, characteristic, complemented
2#26011821C211yesmaximal, characteristic, complemented
3#26011804C211yesmaximal, characteristic, complemented
4#26011791C212yesmaximal, complemented
5#26011706C212yesmaximal, complemented
6#26011642C211yesmaximal, characteristic, complemented
7#26011625C211yesmaximal, characteristic
8#26011601C212yesmaximal, complemented
9#26011591C212yesmaximal, complemented
10#26011548C211yesmaximal, characteristic, complemented
11C2×C4×D4×C173C211yesmaximal, characteristic, complemented
12C2×#14583814C2212yes—
13C2×#14583811C2212yes—
14#19149673—21no—
15#19149671C2211yescharacteristic, complemented
16#19149667C2211yescharacteristic, complemented
17#19149662—21no—
18#19149658C2211yescharacteristic
19#19149653C2211yescharacteristic
20C4×Dic346C2211yescharacteristic, complemented
21C4×D692C2211yescharacteristic, complemented
22C4×C22⋊D173—21no—
23#19149636—21no—
24#19149612C2212yescomplemented
25#19149610C2212yescomplemented
26C4⋊#13063436C2212yescomplemented
27#19149602C2212yescomplemented
28#19149600C2212yes—
29#19149600C2212yes—
30#19149599C2212yescomplemented
31C22⋊#13063436C2211yescharacteristic
32C22⋊#13063436C2211yescharacteristic
33#19149590C2212yescomplemented
34C22×D4×C173C2211yescharacteristic
35C23×C4×C173—21no—
36C4×D4×C173—21no—
37C4×D4×C173—21no—
38C4×D4×C173C2212yes—
39C4×D4×C173C2212yes—
40C2×C4⋊C4×C173C2211yescharacteristic
41C2×C22⋊C4×C173—21no—
42C2×C42×C173C2211yescharacteristic
43C2×Dic346C2311yescharacteristic
44C2×D692C2311yescharacteristic
45C2×C22⋊D173—21no—
46C2×C4×D173—21no—
47#14583814—24no—
48#14583811—24no—
49C2×#13063436C2312yes—
50C2×#13063436C2312yes—
51C4⋊Dic173C2311yescharacteristic
52C4⋊Dic173C2311yescharacteristic
53#14583805—21no—
54#14583805—21no—
55#14583805—21no—
56#14583805—21no—
57C22⋊Dic173—21no—
58#14583803—21no—
59#14583803—21no—
60#14583803—21no—
61#14583803—21no—
62C4×Dic173—21no—
63C24×C173—21no—
64C2×D4×C173—24no—
65C2×D4×C173—21no—
66C2×D4×C173C2312yes—
67C22×C4×C173—21no—
68C22×C4×C173—22no—
69C22×C4×C173—22no—
70C22×C4×C173—44no—
71C22×C4×C173D411yescharacteristic
72C22×C4×C173D411yescharacteristic
73C22×C4×C173C2311yescharacteristic
74C4⋊C4×C173—21no—
75C4⋊C4×C173C2312yes—
76C22⋊C4×C173—22no—
77C22⋊C4×C173—22no—
78C42×C173D411yescharacteristic
79C42×C173D411yescharacteristic
80C42×C173C2311yescharacteristic
81C42×C173C2311yescharacteristic
82C22×D173—21no—
83Dic346—22no—
84D692—22no—
85C22⋊D173—42no—
86C2×Dic173—21no—
87C2×Dic173—21no—
88C2×Dic173—21no—
89C4×D173—42no—
90#13063436—22no—
91#13063436—22no—
92C23×C173—24no—
93C23×C173—22no—
94C23×C173—42no—
95C23×C173—42no—
96C23×C173C2×D411yescharacteristic
97D4×C173—24no—
98D4×C173—22no—
99D4×C173#4114yes—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—41no—
104C2×C4×C173—44no—
105C2×C4×C173—44no—
106C2×C4×C173—42no—
107C2×C4×C173C2×D411yescharacteristic
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2×D411yescharacteristic
111C2×C4×C173C2×D411yescharacteristic
112C2×C4×C173C2411yescharacteristic
113C22×C173—21no—
114C22×C173—21no—
115C22×C173—24no—
116C22×C173—48no—
117C22×C173—42no—
118C22×C173—42no—
119C22×C173AGL(1,ℤ/8)12yes—
120C22×C173C22×D411yescharacteristic
121C2×D173—41no—
122C2×D173—41no—
123C2×D173—41no—
124C4×C173—21no—
125C4×C173—21no—
126C4×C173—21no—
127C4×C173—21no—
128C4×C173—21no—
129C4×C173—42no—
130C4×C173C2×#4112yes—
131Dic173—41no—
132Dic173—41no—
133Dic173—41no—
134C2×C173—22no—
135C2×C173—44no—
136C2×C173#52511yescharacteristic
137C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C173#57511yescharacteristic
139D173—81no—
140C173#311411yescharacteristic, complemented, Sylow
141#3114—1731nomaximal, Sylow
142#529—1731no—
143C2×C4×D4D17311yescharacteristic, complemented
144#483—1731no—
145#480—1731no—
146#478—1732no—
147#463—1732no—
148#450—1731no—
149#447—1731no—
150#442—1732no—
151#440—1732no—
152#432—1731no—
153C22×D4C2×D17311yescharacteristic
154C23×C4—21no—
155C2×SD16—1732no—
156C2×D8—1732no—
157#126—3461no—
158#125—1731no—
159#124—1731no—
160#123—3461no—
161C22⋊Q8—1731no—
162#121—1731no—
163C4×Q8—1731no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—3461no—
167C4×D4—1731no—
168C4×D4C2×D17312yes—
169C4×D4C2×D17312yes—
170#117—3461no—
171C2×C4⋊C4C2×D17311yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D17311yescharacteristic
174#107—1732no—
175#106—1732no—
176C4⋊C8—1732no—
177#103—1732no—
178#102—1732no—
179#102—1732no—
180#102—1732no—
181C22⋊C8—1731no—
182C22⋊C8—1731no—
183#97—1732no—
184C24—21no—
185C2×Q8—1731no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3461no—
189C2×D4—1731no—
190C2×D4C22×D17312yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—3461no—
196C22×C4C22⋊D17311yescharacteristic
197C22×C4C22⋊D17311yescharacteristic
198C22×C4C22×D17311yescharacteristic
199SD16—3464no—
200D8—3464no—
201C2×C8—1732no—
202C2×C8—1732no—
203C4⋊C4—21no—
204C4⋊C4—3461no—
205C4⋊C4—3461no—
206C4⋊C4—3461no—
207C4⋊C4—3461no—
208C4⋊C4—1731no—
209C4⋊C4—1731no—
210C4⋊C4C22×D17312yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—3461no—
214C22⋊C4—3461no—
215C22⋊C4—3461no—
216C22⋊C4—3461no—
217C22⋊C4—3461no—
218C42—3461no—
219C42C22⋊D17311yescharacteristic
220C42C22⋊D17311yescharacteristic
221C42C22×D17311yescharacteristic
222C42C22×D17311yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3461no—
228C23C2×C22⋊D17311yescharacteristic
229Q8—3462no—
230D4—24no—
231D4—22no—
232D4—3462no—
233D4—6922no—
234D4#1458382614yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—3461no—
243C2×C4—3461no—
244C2×C4—3461no—
245C2×C4—6922no—
246C2×C4C2×C22⋊D17311yescharacteristic
247C2×C4C2×C22⋊D17311yescharacteristic
248C2×C4C2×C22⋊D17311yescharacteristic
249C2×C4C2×C22⋊D17311yescharacteristic
250C2×C4C2×C22⋊D17311yescharacteristic
251C2×C4C23×D17311yescharacteristic
252C8—3462no—
253C8—3462no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—6921no—
261C22—6921no—
262C22—6921no—
263C22#1914970512yes—
264C22C22×C22⋊D17311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—6921no—
272C4—6921no—
273C4—6921no—
274C4C2×#1458382612yes—
275C2—22no—
276C2—44no—
277C2—13841no—
278C2#2601200211yescharacteristic
279C2C2×#1914970511yescharacteristic
280C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×C4×D4×C173C2
socleC22×C173C22×D4
prime radical#26011625C2
hypercenterC2×C4×D4D173
hypocenterC173#3114
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26012025C2semidirect
2. #26011821C2semidirect
3. #26011804C2semidirect
4. #26011791C2semidirect
5. #26011706C2semidirect
6. #26011642C2semidirect
8. #26011601C2semidirect
9. #26011591C2semidirect
10. #26011548C2semidirect
11. C2×C4×D4×C173C2semidirect
15. #19149671C22semidirect
16. #19149667C22semidirect
20. C4×Dic346C22semidirect
21. C4×D692C22semidirect
24. #19149612C22semidirect
25. #19149610C22semidirect
26. C4⋊#13063436C22semidirect
27. #19149602C22semidirect
30. #19149599C22semidirect
33. #19149590C22semidirect
140. C173#3114semidirect
143. C2×C4×D4D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260120252
#260118212
#260118042
#260117912
#260117062
#260116422
#260116252
#260116012
#260115912
#260115482
C2×C4×D4×C1732
#3114173